Let \(\tau \subset \sigma \)  be simplices of \(B\) .
By Whitehead's Theorem with twisted coefficients ([1]}, Proposition 4, Chapter 2, Section 3), it's enough to show that the inclusion \(f^{-1} U_\sigma \hookrightarrow f^{-1} U_\tau \)  induces isomorphisms in homotopy groups. We have some immediate observations:

Let \(v\in \tau \) . Then the inclusions
f-1U[rd, hook][r, hook]f-1U[d, hook]
f-1U{v}
commute, so it's enough to consider the case when \(\tau =\lbrace v\rbrace \) .

Let \(b\)  be a point in the interior of \(\sigma \) . Then the inclusions
f-1b[rd, hook][r, hook]f-1U[d, hook]
f-1U
commute, so it's enough to show that the inclusion \(f^{-1}b\hookrightarrow f^{-1}U_\tau \)  induces isomorphisms in homotopy groups.

\(f^{-1}U_{\lbrace v\rbrace }\)  retracts to \(f^{-1}v\)  and \(U_{\lbrace v\rbrace }\)  retracts to \(v\) , by straight-line retractions. Thus from the long exact sequence of the triples \((E,f^{-1}U_{\lbrace v\rbrace },f^{-1}v)\)  and \((B, U_{\lbrace v\rbrace },v)\)  we have that the inclusions \((E,f^{-1} b)\rightarrow (E,f^{-1}U_{\lbrace v\rbrace })\)  and \((B,\lbrace b\rbrace )\rightarrow (B,U_{\lbrace v\rbrace })\)  induce isomorphisms in homotopy groups.

Consider the commutative diagram
(E,f-1 b)[d][r, hook](E,f-1U{v})[d]
(B,{b})[r, hook](B,U{v})
By our third observation, the horizontal maps induce isomorphisms in homotopy groups, and because \(f\)  is a quasifibration the first vertical map induces isomorphisms in homotopy groups as well. Therefore the same holds for the second vertical map.
Now consider the long exact sequences in homotopy of the triples \((E, f^{-1}U_{v}, f^{-1}b)\)  and \((B,U_{\lbrace v\rbrace },b)\) .
[r] i+1(E, f-1b)rd  i+1(E, f-1U{v})r d  i(f-1U{v}, f-1b )d[r]
[r] i+1(B, b)r i+1(B,U{v})r i(U{v}, b)[r]
The first vertical map is an isomorphism because \(f\)  is a quasifibration, and we have shown the second vertical map to be an isomorphism as well. Thus by the Five Lemma the third vertical map is an isomorphism. But \(\pi _{i}(U_{v}, b)=0\) , so we conclude that \( \pi _{i}(f^{-1}U_{\lbrace v\rbrace }, f^{-1}b )=0\) . Thus the inclusion \(f^{-1}b\hookrightarrow f^{-1}U_{\lbrace v\rbrace }\)  induces isomorphisms of homotopy groups
